Advances in Mathematics of Communications (AMC)

Secondary constructions of bent functions and their enforcement

Pages: 305 - 314, Volume 6, Issue 3, August 2012      doi:10.3934/amc.2012.6.305

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Claude Carlet - LAGA, Universities of Paris 8 and Paris 13; CNRS, UMR 7539, Department of Mathematics, University of Paris 8, 2 rue de la liberté, 93526 Saint-Denis cedex 02, France (email)
Fengrong Zhang - School of Computer Science and Technology, China University of Mining and Technology, Xuzhou, Jiangsu 221116, China, and ISN, Xidian University, Xi'an, Shannxi 710071, China (email)
Yupu Hu - State Key Laboratory of Integrated Services Networks, Xidian university, P.O. Box 95, Taibai Road 2, Xi'an, Shannxi 710071, China (email)

Abstract: Thirty years ago, Rothaus introduced the notion of bent function and presented a secondary construction (building new bent functions from already defined ones), which is now called the Rothaus construction. This construction has a strict requirement for its initial functions. In this paper, we first concentrate on the design of the initial functions in the Rothaus construction. We show how to construct Maiorana-McFarland's (M-M) bent functions, which can then be used as initial functions, from Boolean permutations and orthomorphic permutations. We deduce that at least $(2^n!\times 2^{2^n})(2^{2^n}\times2^{2^{n-1}})^2$ bent functions in $2n+2$ variables can be constructed by using Rothaus' construction. In the second part of the note, we present a new secondary construction of bent functions which generalizes the Rothaus construction. This construction requires initial functions with stronger conditions; we give examples of functions satisfying them. Further, we generalize the new secondary construction of bent functions and illustrate it with examples.

Keywords:  Block cipher, stream cipher, Boolean function, bent function, Boolean permutation, orthomorphic permutation.
Mathematics Subject Classification:  06E30, 94A60.

Received: July 2011;      Revised: March 2012;      Available Online: August 2012.